| Author | Message | ||
     Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 30 Registered: 03-2004 |
Is it possible to solve the biharmonic equation for a plate with a completely free edge? Using U: del2(U) = V V: del2(V) = Q where U is the out-of-plane displacement, the sample problems, fixplate.pde and freeplate.pde address two of the three boundary situations encountered in analysis of plate bending - a simply supported edge and a built-in edge. Fixplate: For a built-in edge the boundary conditions are: U = 0 and DU/dn = 0 These are established in fixplate by natural(U) = 0 and natural(V) = L*U, where L is a big number such as 1e4. Freeplate: For a simply supported edge the boundary condtions are: U = 0 and Mn = 0, where Mn is the bending moment. These are established in freeplate by value(U) = 0 and value(V) = 0 Although the second boundary condition is described in freeplate as being approximate, it is actually exact. For example on the boundary parallel with the x-axis, dxx(U) is forced to zero by U = 0. Therefore, Mn = dyy(U) + nu*dxx(U) is reduced to Mn = dyy(U) and value(V) = 0 insures that Mn is exactly 0. So, this works great. Completely free edge: The boundary conditions for a completely free edge are: 1. Mn = 0 2. Myx = 0 3. Qn = 0 , where Mxy is the twisting moment and Qn is the shearing force. On an edge parallel with the x-axis, Mn, Mxy and Qn are defined as: Mn = My = -D*[dyy(U) + nu*dxx(U)] Myx= -D*(1 – nu)*dx(dy(U)) Qn = Qy = dy(My) – dx(Myx) = -D*(dy[dyy(U) + dxx(U)]) Using physical considerations, the 2nd and 3rd conditions can be combined into one. Qy = -dx(Myx) or equivalently dyy(dy(U)) = -(2 – nu)*dy(dxx(U)) Is there a way to do this in FlexPDE? | ||
| Jerry Brown (jerrybrown11743) Member Username: jerrybrown11743 Post Number: 31 Registered: 03-2004 |
There were two errors in the previous post. Although they don't affect the main points, I've corrected them to avoid confusion. Is it possible to solve the biharmonic equation for a plate with a completely free edge? Using U: del2(U) = V V: del2(V) = Q where U is the out-of-plane displacement, the sample problems, fixplate.pde and freeplate.pde address two of the three boundary situations encountered in analysis of plate bending - a simply supported edge and a built-in edge. Fixplate: For a built-in edge the boundary conditions are: U = 0 and DU/dn = 0 These are established in fixplate by natural(U) = 0 and natural(V) = L*U, where L is a big number such as 1e4. Freeplate: For a simply supported edge the boundary condtions are: U = 0 and Mn = 0, where Mn is the bending moment. These are established in freeplate by value(U) = 0 and value(V) = 0 Although the second boundary condition is described in freeplate as being approximate, it is actually exact. For example on the boundary parallel with the x-axis, dxx(U) is forced to zero by U = 0. Therefore, Mn = -D*[dyy(U) + nu*dxx(U)] is reduced to Mn = -D*dyy(U) and value(V) = 0 insures that Mn is exactly 0. So, this works great. Completely free edge: The boundary conditions for a completely free edge are: 1. Mn = 0 2. Myx = 0 3. Qn = 0 , where Mxy is the twisting moment and Qn is the shearing force. On an edge parallel with the x-axis, Mn, Mxy and Qn are defined as: Mn = My = -D*[dyy(U) + nu*dxx(U)] Myx= -D*(1 – nu)*dy(dx(U)) Mxy = D*(1 - nu)*dx(dy(U)) Qn = Qy = dy(My) – dx(Mxy) = -D*(dy[dyy(U) + dxx(U)]) Using physical considerations, the 2nd and 3rd conditions can be combined into one. Qy = -dx(Myx) or equivalently dyy(dy(U)) = -(2 – nu)*dy(dxx(U)) Is there a way to do this in FlexPDE? | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 724 Registered: 06-2003 |
Your last equation could be interpreted as dy[dyy(U) + (2-nu)*dxx(U)]=0 This starts to look a lot like Natural(V)=0, if you can apply some of the same dropouts for properly oriented edges. Any hope there? Some years ago I generated a version of the bending scripts using bending angles and moments explicitly. The moments are the natural BC values. I can't find any notes on the derivation, but maybe you can see what is happening. I have been away from this topic so long, I've forgotten the little I once knew.... { FREEM.PDE } title " Plate Bending - simple support " Select errlim = 0.005 cubic { Use Cubic Basis } Variables Tx,Ty w Definitions xslab = 11.2 yslab = 8 h = 0.0598 {16 ga} L = 1.0e6 E = 29e6 Q = 14.7 nu = .3 G = 0.5*E/(1+nu) D = E*h**3/(12*(1-nu**2)) Mx = D*[dx(Tx) + nu*dy(Ty)] My = D*[nu*dx(Tx) + dy(Ty)] Mxy = D*0.5*(1-nu)*[dy(Tx) + dx(Ty)] beta = 5/6 ! parabolic shear stress distribution in z Sx = beta*G*h*[dx(w)-Tx] Sy = beta*G*h*[dy(w)-Ty] Initial values Tx = 0 Ty = 0 w = 0 Equations Tx: dx(Mx) + dy(Mxy) + Sx = 0 Ty: dy(My) + dx(Mxy) + Sy = 0 w: dx(Sx) + dy(Sy) + Q = 0 Boundaries Region 1 start (0,0) natural(Tx) = 0 natural(Ty) = 0 value(w) = 0 line to (xslab,0) to (xslab,yslab) to (0,yslab) finish monitors contour(w) plots contour(Tx) contour(Ty) contour (w) as "Displacement" elevation(w) from (0,yslab/2) to (xslab,yslab/2) as "Displacement" surface(w) as "Displacement" end 44861 |